Stability of a discrete-time, macroeconomic disequilibrium model

Tilburg University

Stability of a discrete-time, macroeconomic disequilibrium model

Kaper, B.

Publication date:

1982

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Citation for published version (APA):

Kaper, B. (1982). Stability of a discrete-time, macroeconomic disequilibrium model. (pp. 1-13). (Ter Discussie

FEW). Faculteit der Economische Wetenschappen.

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No. 82.18

STABILITY OF A DISCRETE-TIME, MACROECONOMIC DIS-EQUILIBRIUM MODEL.

STABILITY OF A DISCRETE-TIME, MACROECONOMIC DISEQUILIBRIUM MODEL. ABSTRACT.

We invesi:.igate the stability of a macroeconomic monetary discrete-time model

with a constraint on the market for bankcredit. A theorem is proved on

asymp-2

totic stability of a piecewise linear discrete-time system in R which is not

-~-1. INTRODUCTION.

In [3] macroeconomic monetary models have been 3eveloped with constraints on the market for bankcredit. In studying the dynamics of these disequilibrium models we met the problem of asymptotic stability of discrete-time systems which are not standard in the theory of difference equations, [5]. Similar problems arise in continuous-time disequilibrium problems (c.f. for instance [1]). Some advancements on that field of research have been made by e.g.

Laro~ue [4], v.d. Heuvel [2]. In this paper we present a theorem on asymptotic stability of a piecewise linear discrete-time system in ~22 which is not over-all linear. This theorem is applicable to a macroeconomic monetary disequíli-brium model that has been abstracted from [3]. In order to get a connection as well as possible with the disequilibrium models in Koning we will use the

3 -2. A MACRO ECONOMIC DISEQUZLIBRIUM MODEL.

We consider a macroeconomic monetary model with constraints on the market for bankcredit. The variables of the model represent relative deviations of their

values on paths of balanced growth. If x symbolises the actual value and x_e

the balanced-growth value of a variable of the model then the relative de-viation of the actual value with respect to the balanced-gr.owth value of the variable (x) is

X - X

e x

:-xe

In the neighborhood of the equilibrium the first difference (~x),

~x :- x - x-1 1)

is approximately equal to the difference of the actual- and the balanced-growthrate of the variable x in the past period. This is also known as the extra growthrate of the variable in question (cf. the appendix).

If in the model a11 exogeneous variables are zero and there is equality be-tween the actual and balanced-growth value the relative variables of the model will persist in zero-value at all time. If any exogeneous variable is given a non zero value relative variables of the model wi11 leave their zero

posi-tion. In order to analyse the effect of a permanent pulse on any of the exo-geneous variables we will multiply these variables with the Heavisidefunction

H,

H(x) - 0 , if x ~ 0 - 1 , if x ~ 0.

The value of all relative variables of the model will be equal to zero if all exogeneous variables are not effective in the model.

The following quantities are involved by the model (the characters in brac;ket.s represent the exogeneous part of the variable in question):

y national income

c(y) demand for consumption i(t) demand for investment goods

4 -rb interest rate of bankcredit bd demand for bankcredit

b(Q) supply of bankcredit (a: discount rate)

s

The model is given by the following set of equilibrium- and adjustment equa-tions: (2-1) c - Yly t YH (2.2) (2.3) i - -tlrb f -1 y - ~lc t ~2i (2.4) bd - -Slrb f d2y (2.5) bs - Qlrb - aH (2.6) rb - rb f pl (bd - bs ) -1 -1 -1 where b - min {bd, bs} t2(y-y-1) - t3(bd - b-1) t tH -1

'Phe greek characters provided with a subindex are positive (adjustment) con-;;tants of the mode]-. The model will be reduced to a set of first order

dif-2

ference equations in R. From (2.1) -(2.3) we qet an equation for y, ("l.7) _{y - a~ -n2tlrb} -1 where and - n2t2y-1 - n2t3(bd - b-1) f(n1Y t n2t)H] -1 a :- (1-n1Y) - n2t2)

bd - b-1 - min {0, (S1 f crl)rb - d2y-1 - aH}.

-1 -1

From (2.4) -(2.6) we derive an equation for rb,

(2.8) rb - rb t pl(-dlrb _{} d2y-1} - alrb t aH).

-1 -1 -1

Let us introduce a new set of variables, xl(n) :- Y-1

x2 (n) -- rb

S -We get a first-order system in ~2z:

xl(nt1) --an2t2 x1(n) - ar)2t1 x2(n) - ar12t3 min {0,

(2.9) -bZxl(n) f(dl f al)x2(n) - QH(n)} t a(rl1Y t n2t)H(n) x2(nfl) - Pld2 xl(n) f(1-81P1 - a1P1)x2(n) f P1aH(n).

Assuming that all variables were equal to zero up to the zeroth period we

get at period 1

~xl (1) - -arl2t3 min {0, -~}

~,x2(1) - PiQ

This will be taken as the initial value of the first order system ( 2.9). The

Heaviside function then might be replaced by the value 1. System ( 2.9) con-sists of two linear subsystems whereas the system itself is ofcourse not overall linear:

3ubsystem 1, if -d2xi(n) f(dl t ol)x2(n) - a ~ 0 x(nfl) - Alx(n) t bi

where

A1 - -a~2t2 t arl2t3ó2 -an2t1 - ar12t3(Slfai)

p1cS2 1-(81fQ1)K~1 and T bl -[ an2t3a t a ;n1Y f p~t ) , c~1Q~ T x(n) :- [xl(n), x2(n~l ; Subsystem 2, if -82x1(n) t(dltal)x2(n) - a~ 0 x(nfl) - A2x(n) t b2 where

A`' - -arl~ ~ ~G -ar~2 t 1

p S~

1 1 - (81t61)P1

6

-b2 - ~a(n1Y f n~t). Pla] .

The equilibrium position x of the overall system (2.9) is just equal to the common equilibrium solution of both subsystems provided it exists (c.f, the appendix),

x-(I - A.)-lb, i- 1 or 2.

- i

-Firially we apply a linear transformation to system (2.9), y(ni - x(n1 - x,

which transforms (2.9) into a homogeneous first-order system in R2,

(ntl) - -an2t2y1(n) - a~2tly2(n) - min {0. -S2y1(n~ f (Slfal)y2(n)} (2.10)

2(nfl) - c~1cS2y~(n) t (1-S1P1-Qlpl)y2(n) or equivalently into two linear subsystems,

if

-d2y1 (n) f (dltal)Y2(n) ~ 0 ~(nfl) - Aly(n),

else

~(ntl) - A2y(n) .

The minimum function in (2.10) implies continuity of the right hand side of (2.10). In the next section it will be shown that the 0 solution of such a

2

homogeneous pieccwi:;e linear system in ik will be asymptotically stable if both subsystems art asymptotically stable.

CONCLUSION: The equilibrium of (2.10) is asymptotically stable if both sub-systems are asymptotically stable, i.e. if

Itr A,I-1 ~ det A, ~ 1, i E{1,2}

i i

7

-(2.11) ~-an7t2 f aP2t.1S~ t 1-(dltol)P1~ _{- 1~(-a~2t2 f an2t3ó2)(1-[ó1tQ1~P1)}

t ar~2t 3~ FI-~ol) ) P1S~ ~ 1

(l. l~) ~-ar~ltG t 1-(61to1 ) I~1 ~- 1 ~(-a~2t2) ( 1 -[ 61}~1~ p1 ) f an2tlE~lu2 ~ 1 NUMERICAL EXAMPLE: If we take the following set of coefficients in the ~nodel

(2.1) -(2.6) then tYie conditions (2.11) and (2.12) are satisfied and the mo-del is asymptotically stable:

f3

-2

3. PIECEWISE-LINEAR DISCRETE DYNAMIC SYSTEMS IN R.

We ccnsei~.lr~r tlr~~ lcl l.uwinc~ autonomous i~iecewise-linear diiYerence equation in

2 R (3.1) _{~tl - Axn} where A:- A, if x E C., i E I .- {1,2,...,n}, i - i n

C, be closed cones in R2 with vertices in the origin, with disjoint

1 n

interiors, U C. -~t2; Let the numbering of the cones around the i-1 1

origin be anticlockwise Ci ~~ Ci}1 '- {a

~ifllu~i.f11~ - 1~ a z 0}, i E In, ntl :- 1

We will prove asymptotic stability of the zero s~~lution of system (3.1) if 0 is an asymptotically stable solution of each of the linear subsystems

x - A.x

-~fl i~

and the function in the right hand side of (3.1) is continuous. THEOREM. Consider system (3.1). If (3.2) I tr A. I- 1 ~ det A, ~ 1, Ki E I, i i n and (3.3) Ai ~i - Ai-1 ~

then system (3.1) is asymptotically stable.

PROOF. We will make use of the concept of a Liapunov function (c.f. [5]). The form of the Liapunov function is based on the one constructed by Laroque for piecewise linear differential systems (c.f. [4]).

A refinement of Laroque's function has been introduced by van den Heuvel in 2

his thesis ( 2] . Let V: R ~ R, defined by

V(x) :- det2(x, Ax] t tlixll4

9

-Then

~I(x) - V(Ax) - V(x).

V(x )- det2[ Ax, A2x] t eU Axll 4- detZ[ x, Ax] - ell xll 4

1) {det2(Aj) - 1} det2[ x, Ax] t e{IIAxU4 - Uxll4} 2

The expression det [x, Ax] equals zero if x is a real eigenvector of A. Con-sider the case that for some index i E I_n A. has an eigenvalue_i with a real eigenvector y that belongs to Int(Ci). By virtue of (3.2) we have ~~~ ~ 1. Choose Y such that 0 ~ Y ~ 1- a4. Then the set K defined by

Y

Ky :- {x E R2~IIAxll4 - Ilxll4 ~-yllxll4} rl Int (Ci)

contains y. Ky is an open cone. If the eigenvector of A, just equals q, theii by

i --i

the continuity property of A~ is also an eigenvector of Ai1~qi Ai ~i

-Ai-1 ~~

In that case in the definition of K the intersection should be taken with qi

Int (C,_i-1 U C ). i

Let V be the set of all eigenvectors of A satisfying the above conditions,

v:- {y I H i E In:[ AiY - ay, a E IR,y ~ OJ nj y E

ci~ }.

To each ~ E V an open cone K can be assigned. Define Y

K :- U K .

yE V y

Then by assumption (3.2) and the definition of K we have for each x E K (3.4) V(x) ~ -FyUxu4.

Let us determine next the scalars a, f3, and e a:- min {det2[ x, Ax] I U xll - 1, x~ K}

(3 :- max {IlAxll

~ IIxU - 1}

- 10 -and

a 2

(3.5) ~:- 2S4 [ 1 - max idet (Ai) I i E In}] ,

Note that the scalars a and t3 exist by the continuity of the respective functions on compact sets. For each x~ K we have

V(x) ~{det2(A~) - 1}aIIxB4 t 2S4( i - max {det2(Ai) I i E In}). ,~411x114.

and hence

(3.6) V(x) ~ laidet2(A.) - 1}IIxq4,

- - 2 ~

-From the definition of V and the inequalities (3.4) and (3.6) we may conclude that V is a Liapunov function in the classical case: V(x) and -V(x) are

11 -Appenr~ices.

1. Let us denote the actual and balanced growth rate in the past period by g repectively g . Then

e

x- ( ltg) x-1 and xe -( 1 fge) xe

-1

The first difference of the relative variable x is

x-xe ~-1-xe-1 _{X-1 1}

!~x - ~ - X _{- - Q--- .}

1 f . ( g-ge ) „ g-4e .

e

e-1 e-1 ge

2. The equilibrium positions of the linear subsystems are equivalent if there holds the relation

(a.l) (IAe1)lbl

-(I-A2)-lb2

Let us define A, b, where A1 - A2 f A and bl - bz f b,

A - an213S2 -an213(dl}vl)

and

0 0

b - [ar12t3o, 0]T.

Relation (a.l) will successively be reduced into the following relations: bZ ~- b - (I-A2-A)(I-A2)-lb2

or eventually b

--A(I-A2)-1b2.

The last relation can be checked by straiqht forward substitution.

T 3. The equilibrium position is given by x-[xl, x2] ,

12

-R-(1 t an2t2)(d1fQi)pl t an2tlpld2.

13

-~1~ E~kalbar J.C., The stability of non-Walrasian processes, Ecvnometrica 48 (1980), 371-386.

~2~ FIeuvel, P. van den, The stability of a macroeconomic system with quanti-ty constraints (1981), Thesis 'Pechnische Hogeschool, Eindhoven, the Netherlands.

[3) Koning, J.H., Kredietrantsoenering en onevenwichtigheid (1982), Thesis, Tilburg University, the Netherlands.

~4~ C,aroque, G., Notes and Comments, A comment on "Strahle Spillover amonc~ Subst.i.tut:es", Iteview of Economic Studies (19fi1), xz VII7, 35'~-361. [5] Lasalle, J.P., Stabi.lity theory for difference equations, Studies in

- 14 -IN 1981 REEDS VERSCHENEN:

0.1. J.J.A. Moors _{Inadmissibility of linearly}

invariant estimators in

truncated parameter spaces jan.

0.2. H. Peer De mathematische structuur

J. Klijnen van

conjunctuur-structuur-modellen en een rekenprocedure

voor numerieke simulatie van deze modellen

Definities van gemiddelde factor-productiviteiten en bezettings-graad in een jaargangenmodel voor industriële sectoren, met een toepassing voor de sector Chemische Industrie

0.3. H. Peer _{Macro economic policy options in}

non-markt structures febr.

0.4. J. van Mier ~-vergelijkinger en operatoren maart

0.5. A.L. Hempenius 0.6. R.J.M. Heuts 0.7. B. Kaper 0.8. R.M.J. Heuts and R. Willemse 0.9. J.P. Heesters 10. J.P. Heesters

Asymptotic Robustness of Prediction Intervals of Arima Models by Devia-tions of Normality

Some aspects of differential

equa-tions with discontinuous right-hand

sides

jan.

maart

mei

juli

Impulse response patterns for various

dynamic time teries models juni

Aankleden of uitkleden?

Een kritische beschouwing van de honorering van de huisarts - vrij beroepsbeoefenaar

Aankleden of uitkleden?

Een kritische beschouwing van de honorering van de medisch specia-list - vrij beroepsbeoefenaar ten opzichte van de ambtenaar

sept.

okt. 11. Dr. G.P.L van Roij Rente-arbitrage, valutaspeculatie

en wisselkoersen nov.

12. J. Glombowski _{A Comment on Sherman's Marxist}

Cycle Model

revised version

13. Drs. W.A.M. de Lange _{Deeltijdarbeid op de Katholieke} H.A.C. de Coninck-Merckx Hogeschool Tilburg

M.R.M. Turlinas M.C.M. Puyk

nov.

15

-14. Drs, w.A.M. de Lange _{Tabellenboek bij het Onderzoek}

L.H.M. Bosch _{'Deeltijdarbeid op de Katholieke}

M.C.M. Turlings _{Hogeschool Tilburg' nov.}

15. H. Peer _{Economische groei en uitputtelijke}

IN 1982 REEDS VERSCHENEN: O1. W. van Groenendaal

02. M.D. Merbis 03. F. Boekema 04. P.T.W.M. Veugelers 05. F. Boekema 06. P. van Geel 07. J.H.M. Donaers, F.A.M, van der Reep

08. R.M.J. Heuts

09. B.B. van der Genugten

10. J. Roemen 11. J. Roemen 12. M.D. Merbis 13. P. Slangen 14. M.D. Merbis - 16

-Building and analyzing an jan

econometric model with the use of a hybrid computer;

part I.

System properties of the jan.

interplay model

Decentralisatie en regionaal maart sociaal-economisch beleid

Een monetaristisch model voor maart de Nederlandse economie

Morfologie van de "Wolstad", april Over het ontstaan en de

ont-wikkeling van de ruimtelijke geleding en struktuur van Tilburg.

Over de (on)mogelijkheden mei

van het model van Knoester.

De betekenis van het monetaire

beleid voor de Nederla.~dse

ecc-nomie, presentatie van een ana-lyse aan de hand van een

een-voudig model

The use of non-linear trans-formation in ARIMA-MOdels when the data are non-Gaussian

distributed

mei

juni Asymptotic normality of least

squares estimators ín auto-regressive linear regression

models. j~i

Van koetjes en kalfjes Z juli

Van koetjes en kalfjes II juli

On the compensator Part I

Problem formulation and

prelimi-naries juli

Bepaling van de optimale beleids-parameters voor een stochastisch kasbcheersprobleem met continue

controle aug.

Linear - Quadratic - Gaussian

- 17

-15. P. Hinssen _{Een kasbeheermodel onder}

J. Kriens _onzekerheid

sept. J. Th. van Lieshout

16. A. Hendriks en _{"Van Bedrijfsverzamelgebouw}

T, van der Bij-Veenstra _{naar Bedrijvencentrum" okt.}

17. F.W.M. Boekema _{Industriepolitiek, Regionaal}

A.J. Hendriks _{beleid en Innovatie okt.}